Find a Splice Site in a DNA Sequence 

A DNA sequence consists of base letters A, C, G, and T. Suppose there is a sequence that begins in an exon, contains a splice site, and ends in an intron. If the exons have a uniform base composition, the introns are deficient in C and G, and the splice site consensus nucleotide is G with probability 0.95, the frequency distributions are as follows.

In[1]:=

X exon = {0.25, 0.25, 0.25, 0.25}; intron = {0.4, 0.1, 0.1, 0.4}; splice = {0.05, 0, 0.95, 0};

In[2]:=

X dnaSeq = Characters["CTTCATGTGAAAGCAGACGTAAGTCA"] /. {"A" -> 1, "C" -> 2, "G" -> 3, "T" -> 4};

The state machine has states for exon (1), splice (2), intron (3), and end (4), with the following transition probabilities between states.

In[3]:=

X tm = {{0.9, 0.1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0.9, 0.1}, {0, 0, 0, 1}};

The emissions are nucleotides A (1), C (2), G (3), T (4), or end (5).

In[4]:=

X em = PadRight[{exon, splice, intron, UnitVector[5, 5]}];

In[5]:=

X hmm = HiddenMarkovProcess[1, tm, em];

Find the most probable nucleotide subsequence (exon, splice, intron, or end).

In[6]:=

X sites = FindHiddenMarkovStates[Append[dnaSeq, 5], hmm]

Out[6]=

Find the joint probability of the preceding nucleotide sequence and the DNA sequence.

In[7]:=

X Fold[Times, Likelihood[DiscreteMarkovProcess[hmm], sites], MapThread[Part[em, #1, #2] &, {sites, Append[dnaSeq, 5]}]]

Out[7]=